# Quotient by subgroup is well defined if and only if subgroup is normal

Let $$G$$ be a group and $$N$$ a normal subgroup of $$G$$. Then we may define the quotient group $$G/N$$ to be the set of left cosets $$gN$$ of $$N$$ in $$G$$, with the group operation that $$gN + hN = (gh)N$$. This is well-defined if and only if $$N$$ is normal.

# Proof

## $$N$$ normal implies $$G/N$$ well-defined

Recall that $$G/N$$ is well-defined if “it doesn’t matter which way we represent a coset”: whichever coset representatives we use, we get the same answer.

Suppose $$N$$ is a normal subgroup of $$G$$. We need to show that given two representatives $$g_1 N = g_2 N$$ of a coset, and given representatives $$h_1 N = h_2 N$$ of another coset, that $$(g_1 h_1) N = (g_2 h_2)N$$.

So given an element of $$g_1 h_1 N$$, we need to show it is in $$g_2 h_2 N$$, and vice versa.

Let $$g_1 h_1 n \in g_1 h_1 N$$; we need to show that $$h_2^{-1} g_2^{-1} g_1 h_1 n \in N$$, or equivalently that $$h_2^{-1} g_2^{-1} g_1 h_1 \in N$$.

But $$g_2^{-1} g_1 \in N$$ because $$g_1 N = g_2 N$$; let $$g_2^{-1} g_1 = m$$. Similarly $$h_2^{-1} h_1 \in N$$ because $$h_1 N = h_2 N$$; let $$h_2^{-1} h_1 = p$$.

Then we need to show that $$h_2^{-1} m h_1 \in N$$, or equivalently that $$p h_1^{-1} m h_1 \in N$$.

Since $$N$$ is closed under conjugation and $$m \in N$$, we must have that $$h_1^{-1} m h_1 \in N$$; and since $$p \in N$$ and $$N$$ is closed under multiplication, we must have $$p h_1^{-1} m h_1 \in N$$ as required.

## $$G/N$$ well-defined implies $$N$$ normal

Fix $$h \in G$$, and consider $$hnh^{-1} N + hN$$. Since the quotient is well-defined, this is $$(hnh^{-1}h) N$$, which is $$hnN$$ or $$hN$$ (since $$nN = N$$, because $$N$$ is a subgroup of $$G$$ and hence is closed under the group operation). But that means $$hnh^{-1}N$$ is the identity element of the quotient group, since when we added it to $$hN$$ we obtained $$hN$$ itself.

That is, $$hnh^{-1}N = N$$. Therefore $$hnh^{-1} \in N$$.

Since this reasoning works for any $$h \in G$$, it follows that $$N$$ is closed under conjugation by elements of $$G$$, and hence is normal.

Parents:

• Normal subgroup

Normal subgroups are subgroups which are in some sense “the same from all points of view”.